$C$-differentials, multiplicative uniformity and (almost) perfect $c$-nonlinearity

TL;DR

This paper introduces a new multiplicative differential concept and $c$-differential uniformity, revealing the existence of perfect $c$-nonlinear functions even in characteristic 2 and analyzing their properties and implications for cryptography.

Contribution

It defines $c$-differential uniformity, characterizes it via Walsh transform, and explores perfect $c$-nonlinear functions, including their existence and behavior in cryptographic functions.

Findings

Existence of perfect $c$-nonlinear functions in characteristic 2.

Characterization of $c$-differential uniformity via Walsh transform.

Analysis of $c$-differential uniformity for inverse functions in cryptography.

Abstract

In this paper we define a new (output) multiplicative differential, and the corresponding $c$-differential uniformity. With this new concept, even for characteristic $2$, there are perfect $c$-nonlinear (PcN) functions. We first characterize the $c$-differential uniformity of a function in terms of its Walsh transform. We further look at some of the known perfect nonlinear (PN) and show that only one remains a PcN function, under a different condition on the parameters. In fact, the $p$-ary Gold PN function increases its $c$-differential uniformity significantly, under some conditions on the parameters. We then precisely characterize the $c$-differential uniformity of the inverse function (in any dimension and characteristic), relevant for the Rijndael (and Advanced Encryption Standard) block cipher.